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🟦 Math & Geometry: Pow(x, n)

📝 Problem Description

Implement pow(x, n), which calculates x raised to the power n (i.e., \(x^n\)).

Real-World Application

This algorithm is essential for public-key cryptography (RSA exponentiation), computational geometry (distance metrics), and scientific simulations where rapid power calculation is critical for performance.

🛠️ Constraints & Edge Cases

  • \(-100.0 < x < 100.0\)
  • \(-2^{31} \le n \le 2^{31}-1\)
  • Edge Cases: \(n = 0\), \(n < 0\) (reciprocal).

🧠 Approach & Intuition

The Aha! Moment

Exponentiation can be solved in logarithmic time using "Binary Exponentiation" (Exponentiation by Squaring): \(x^n = (x^{n/2})^2\) if \(n\) is even.

🐢 Brute Force (Naive)

Multiply \(x\) by itself \(n\) times. \(\mathcal{O}(n)\), which will time out for large \(n\).

🐇 Optimal Approach

Use recursive or iterative approach: 1. If \(n\) is even, \(x^n = (x^2)^{n/2}\). 2. If \(n\) is odd, \(x^n = x * (x^2)^{(n-1)/2}\). 3. Handle negative \(n\) by \(x^n = (1/x)^{-n}\).

🧩 Visual Tracing

graph LR
    A[x^8] --> B[x^4 * x^4]
    B --> C[x^2 * x^2]
    C --> D[x * x]

💻 Solution Implementation

(Implementation details need to be added...)

⏱️ Complexity Analysis

  • Time Complexity: \(\mathcal{O}(\log N)\) because the exponent is halved in each step.
  • Space Complexity: \(\mathcal{O}(\log N)\) (recursion stack) or \(\mathcal{O}(1)\) (iterative).

🎤 Interview Toolkit

  • Harder Variant: Implement modulo power (x^n) % m.
  • Alternative Data Structures: Iterative version is preferred to avoid recursion depth issues.
  • [Missing Number](#) — Using math to find values.
  • [Sqrt(x)](#) — Other math optimization problems.